A linear homogeneous second order differential equation is of the form:
There will be two solution to the above differential equation:
And all general solutions will be of the form:
The method of Reduction of order
assumes that one solution is already know (i.e. we know that
is a solution the above differential equation) and uses a specific method to find another solution.
We will use interchangeably
, and other shorthand notation of that form.
Steps to solve Reduction of Order problems:
1) Assume that your second solution is of the form
is the solution we know.
4) Get rid of the solution with
, and rearrange your equation to isolate
6) Solve for
in the new first order differential equation
back in and then solve for
8) The second solution to the differential equation is
, with the
found from the previous steps.